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(Solved): (a) For a positive, decreasing function f:[1,)R that is integrable on e ...



(a) For a positive, decreasing function \( f:[1, \infty) \rightarrow \mathbb{R} \) that is integrable on every \( [a, b] \sub???????

(a) For a positive, decreasing function that is integrable on every , define Show that is nonincreasing, bounded, and . (Hint: note that (b) Recall that the harmonic numbers are defined Show that Euler's constant exists, and that . (Hint: use part (a) with .)


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5.a) To show that {a_n} is non-increasing, we can show that for all n, an?an+1. We have:an=?k=1nf(k)??1nf(x)dxan+1=?k=1n+1f(k)??1n+1f(x)dxNow, by subt
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